Problem rigging with automatic weights and shape keys for Unity
Have you tried using a Driver for that?


Add a bone to your armature that will be your Driver, it works
similar to an IK bone, as in, it doesn't affect the mesh directly.
Select the shapekey you want to add the driver to, click with the
right mouse button on the bar below and select Add Driver, it shall
turn purple.
Then on the Graph editor window, where it says F-Curve, change it to
Drivers. Click on "Key", there you can select each shapekey that will
be controlled by the Driver.
Expand the right panel, there you can select the bone that will
control the shapekey, a.k.a. the Driver.
One detail though, On the right panel where it says Type, change it
to Average Value, then moving the bone will change the values on the
shapekey.


There rest is up to your preferences.

Hope it helped.